Time Value of Money

The time value of money (TVM) is arguably the most fundamental concept in finance. It states that a dollar today is worth more than a dollar tomorrow because money available now can be invested to earn returns. This principle underlies virtually all financial decision-making, from personal investments to corporate capital budgeting.

Future Value

If you invest an amount today, how much will it grow to in the future? The future value (FV) answers this question.

Simple Interest

With simple interest, you earn returns only on the principal: $FV = PV \times (1 + r \times t)$

where $PV$ is present value, $r$ is the interest rate per period, and $t$ is the number of periods.

Compound Interest

With compound interest, you earn returns on both principal and accumulated interest: $FV = PV \times (1 + r)^n$

where $n$ is the number of compounding periods.

Example: Investing 1,000 at 8% annually for 10 years: $$FV = 1000 \times (1.08)^{10} = \2,158.92$$

Continuous Compounding

As compounding frequency approaches infinity, we get continuous compounding: $FV = PV \times e^{rt}$

This formula is extensively used in derivatives pricing and theoretical finance.

Present Value

Present value (PV) is the inverse operation: what is a future cash flow worth today?

$PV = \frac{FV}{(1 + r)^n}$

The process of calculating present value is called discounting, and $r$ is the discount rate.

Discount Factors

The discount factor $DF$ represents the present value of $1 received at time$t$: $DF_t = \frac{1}{(1 + r)^t}$

For a series of cash flows, the present value is: $PV = \sum_{t=1}^{n} CF_t \times DF_t = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t}$

Annuities

An annuity is a series of equal payments at regular intervals.

Ordinary Annuity (Payments at End of Period)

Present value: $PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$

Future value: $FV = PMT \times \frac{(1 + r)^n - 1}{r}$

Annuity Due (Payments at Beginning of Period)

Multiply ordinary annuity formulas by $(1 + r)$: $PV_{due} = PV_{ordinary} \times (1 + r)$

Perpetuity

A perpetuity is an annuity that continues forever. Its present value simplifies elegantly: $PV = \frac{PMT}{r}$

For a growing perpetuity where payments grow at rate $g$: $PV = \frac{PMT}{r - g}$

This formula is the foundation of the Gordon Growth Model for stock valuation.

Net Present Value (NPV)

NPV is the cornerstone of investment analysis. It sums all present values of cash flows, including the initial investment (typically negative):

$NPV = -C_0 + \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t}$

Decision rule: Accept projects with $NPV > 0$; reject those with $NPV < 0$.

Internal Rate of Return (IRR)

The IRR is the discount rate that makes $NPV = 0$: $0 = -C_0 + \sum_{t=1}^{n} \frac{CF_t}{(1 + IRR)^t}$

The IRR must typically be solved numerically (using Newton-Raphson or similar methods).

Decision rule: Accept if $IRR >$ required return.

Caution: IRR can give misleading results with non-conventional cash flows or when comparing mutually exclusive projects.

Effective Annual Rate (EAR)

When interest is compounded more frequently than annually, the stated rate differs from the effective rate:

$EAR = \left(1 + \frac{r_{nominal}}{m}\right)^m - 1$

where $m$ is the number of compounding periods per year.

Example: A 12% nominal rate compounded monthly: $EAR = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 12.68\%$

Applications in Programming

TVM calculations are fundamental to financial software:

1. Loan Amortization: Calculate monthly payments and build amortization schedules
2. Bond Pricing: Discount future coupon payments and principal
3. Project Valuation: Implement NPV and IRR functions
4. Retirement Planning: Model savings growth and withdrawal strategies

Efficient implementations require:

• Handling edge cases (zero rates, infinite periods)
• Numerical methods for IRR calculation
• Precision considerations for large time horizons

Choosing the Discount Rate

The appropriate discount rate depends on the context:

• Risk-free rate: Government bond yields (for riskless cash flows)
• Cost of capital: For corporate investment decisions
• Required return: Incorporating risk premiums for risky assets

The discount rate should reflect the opportunity cost of capital: what return could be earned on an investment of similar risk.